Theorem onsucuni2 3914

Description: A successor ordinal is the successor of its union. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
onsucuni2	|- ((A e. On /\ A = suc B) -> suc U.A = A)

Proof of Theorem onsucuni2

Step	Hyp	Ref	Expression
1		unieq 3185	. . . . . 6 |- (A = suc B -> U.A = U.suc B)
2		suceq 3729	. . . . . 6 |- (U.A = U.suc B -> suc U.A = suc U.suc B)
3	1, 2	syl 12	. . . . 5 |- (A = suc B -> suc U.A = suc U.suc B)
4		suceq 3729	. . . . . 6 |- (A = suc B -> suc A = suc suc B)
5	4	unieqd 3188	. . . . 5 |- (A = suc B -> U.suc A = U.suc suc B)
6	3, 5	eqeq12d 1899	. . . 4 |- (A = suc B -> (suc U.A = U.suc A <-> suc U.suc B = U.suc suc B))
7		eleq1 1957	. . . . . 6 |- (A = suc B -> (A e. On <-> suc B e. On))
8	7	biimpac 462	. . . . 5 |- ((A e. On /\ A = suc B) -> suc B e. On)
9		eloni 3667	. . . . 5 |- (suc B e. On -> Ord suc B)
10		ordsuc 3895	. . . . . . . 8 |- (Ord B <-> Ord suc B)
11		ordunisuc 3911	. . . . . . . 8 |- (Ord B -> U.suc B = B)
12	10, 11	sylbir 218	. . . . . . 7 |- (Ord suc B -> U.suc B = B)
13		suceq 3729	. . . . . . 7 |- (U.suc B = B -> suc U.suc B = suc B)
14	12, 13	syl 12	. . . . . 6 |- (Ord suc B -> suc U.suc B = suc B)
15		ordunisuc 3911	. . . . . 6 |- (Ord suc B -> U.suc suc B = suc B)
16	14, 15	eqtr4d 1928	. . . . 5 |- (Ord suc B -> suc U.suc B = U.suc suc B)
17	8, 9, 16	3syl 24	. . . 4 |- ((A e. On /\ A = suc B) -> suc U.suc B = U.suc suc B)
18	6, 17	syl5bir 227	. . 3 |- (A = suc B -> ((A e. On /\ A = suc B) -> suc U.A = U.suc A))
19	18	anabsi7 555	. 2 |- ((A e. On /\ A = suc B) -> suc U.A = U.suc A)
20		eloni 3667	. . . 4 |- (A e. On -> Ord A)
21		ordunisuc 3911	. . . 4 |- (Ord A -> U.suc A = A)
22	20, 21	syl 12	. . 3 |- (A e. On -> U.suc A = A)
23	22	adantr 425	. 2 |- ((A e. On /\ A = suc B) -> U.suc A = A)
24	19, 23	eqtrd 1925	1 |- ((A e. On /\ A = suc B) -> suc U.A = A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  U.cuni 3177  Ord word 3656  Oncon0 3657  suc csuc 3659

This theorem is referenced by:  rankxplim3 5825  rankxpsuc 5826

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663

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